Quasirandom Ramblings

In the early 1990s Spassimir Paskov, then a graduate student at Columbia University, began analyzing an exotic financial instrument called a collateralized mortgage obligation, or CMO, issued by the investment bank Goldman Sachs. The aim was to estimate the current value of the CMO, based on the potential future cash flow from thousands of 30-year mortgages. This task wasn’t just a matter of applying the standard formula for compound interest. Many home mortgages are paid off early when the home is sold or refinanced; some loans go into default; interest rates rise and fall. Thus the present value of a 30-year CMO depends on 360 uncertain and interdependent monthly cash flows. The task amounts to evaluating an integral in 360-dimensional space.

There was no hope of finding an exact solution. Paskov and his adviser, Joseph Traub, decided to try a somewhat obscure approximation technique called the quasi–Monte Carlo method. An ordinary Monte Carlo evaluation takes random samples from the set of all possible solutions. The quasi variant does a different kind of sampling—not quite random but not quite regular either. Paskov and Traub found that some of their quasi–Monte Carlo programs worked far better and faster than the traditional technique. Their discovery would allow a banker or investor to assess the value of a CMO with just a few minutes of computation, instead of several hours.

It would make a fine story if I could now report that the subsequent period of “irrational exuberance” in the financial markets—the frenzy of trading in complex derivatives, and the sad sequel of crisis, collapse, recession, unemployment—could all be traced back to a mathematical innovation in the evaluation of high-dimensional integrals. But it’s just not so; there were other causes of that folly.

On the other hand, the work of Paskov and Traub did have an effect: It brought a dramatic revival of interest in quasi–Monte Carlo. Earlier theoretical results had suggested that quasi–Monte Carlo models would begin to run out of steam when the number of dimensions exceeded 10 or 20, and certainly long before it reached 360. Thus the success of the experiment was a surprise, which mathematicians have scrambled to explain. A key question is whether the same approach will work for other problems.

The whole affair highlights the curiously ambivalent role of randomness in computing. Algorithms are by nature strictly deterministic, yet many of them seem to benefit from a small admixture of randomness—an opportunity, every now and then, to make a choice by flipping a coin. In practice, however, the random numbers supplied to computer programs are almost never truly random. They are pseudorandom—artful fakes, meant to look random and pass statistical tests, but coming from a deterministic source. What’s intriguing is that the phony random numbers seem to work perfectly well, at least for most tasks.

Quasirandom numbers take the charade a step further. They don’t even make the effort to dress up and look random. Yet they too seem to be highly effective in many places where randomness is called for. They may even outperform their pseudorandom cousins in certain circumstances.